The strong Legendre condition and the well-posedness of mixed Robin problems on manifolds with bounded geometry

Let $M$ be a smooth manifold with boundary $partial M$ and bounded geometry, $partial_D M subset partial M$ be an open and closed subset, $P$ be a second order differential operator on $M$, and $b$ be a first order differential operator on $partial M smallsetminus partial_D M$. We prove the regularity and well-posedness of the mixed Robin boundary value problem $$Pu = f mbox{ in } M, u = 0 mbox{ on } partial_D M, partial^P_ u u + bu = 0 mbox{ on } partial M setminus partial_D M$$ under some natural assumptions. Our operators act on sections of a vector bundle $E to M$ with bounded geometry. Our well-posedness result is in the Sobolev spaces $H^s(M; E)$, $s geq 0$. The main novelty of our results is that they are formulated on a non-compact manifold. We include also some extensions of our main result in different directions. First, the finite width assumption is required for the Poincar{e} inequality on manifolds with bounded geometry, a result for which we give a new, more general proof. Second, we consider also the case when we have a decomposition of the vector bundle $E$ (instead of a decomposition of the boundary). Third, we also consider operators with non-smooth coefficients, but, in this case, we need to limit the range of $s$. Finally, we also consider the case of uniformly strongly elliptic operators. In this case, we introduce a emph{uniform Agmon condition} and show that it is equivalent to the Gaa rding inequality. This extends an important result of Agmon (1958).